Cosmology Lecture 01

The Cosmological Principle

The Cosmological Principle states that the universe is spatially isotropic (looks the same in all directions) and homogeneous (has constant density everywhere) on large scales. The Perfect Cosmological Principle states that the universe is also temporally isotropic and homogeneous (a steady state universe). This is unlikely because it doesn’t describe the Cosmic Microwave Background (CMB). The CMB and Hubble’s Law are both provide evidence for isotropy and homogeneity.

Hubble’s Law (1929)

Hubble’s Law is an empirical law stating that, on large scales, recessional velocity is proportional to distance from observer.

${v=Hr}\,\!$

where $H$, the Hubble parameter, is not constant, but can vary slowly with time. By convention, $H$ is often expressed as $H=100\cdot h{km \over s\cdot Mpc}$, where 1 parsec (pc) $\approx 3\cdot 10^{18}cm=3.26ly$, is the distance at which 1 AU appears as 1 arcsec on the sky. The Hubble Space Telescope Key Project (Freedman et al. ApJ 553, 47, 2001) measured the present day value of Hubble Constant $H_{0}=72\pm 8{km \over s\cdot Mpc}$, giving us that the current timescale for the expansion of the universe is $H_{0}^{-1}\approx {h \over 10^{11}}yrs\approx 9.778h^{-1}Gyrs$.

The Scale Factor $a(t)$

$a(t)$ relates physical ($r$) and comoving ($x$) coordinates in an expanding universe:

${\begin{aligned}r&=a(t)x\\{\dot {r}}&={\dot {a}}x+a{\dot {x}}=\underbrace {{\dot {a}} \over a} _{\equiv H}r+\underbrace {a{\dot {x}}} _{\equiv v_{p}}\\\end{aligned}}\,\!$

Thus, the two components of physical velocity are $H$ (the Hubble expansion parameter) and $v_{p}$ (the peculiar velocity, or motion relative to expansion) By convention, $t_{0}\equiv$ today and $a(t_{0})=1$.

The Friedmann Equations

The Friedmann Equation is an equation of motion for $a(t)$. A rigorous derivation requires General Relativity, but we can fake it with a quasi-Newtonian derivation. We will model the universe as an adiabatically ($\Delta S=0$) expanding, isotropic, homogeneous medium. Isotropy allows us to use $r$ as a scalar. Consider a thin, expanding spherical shell of radius $a$. Birkhoff’s Theorum states that even in GR, the motion of such a shell depends only on the enclosed mass $M={4\pi \over 3}a^{3}\rho$. Thus, the energy per unit mass per unit length is:

$E=\overbrace {{\frac {1}{2}}{\dot {a}}^{2}} ^{Kinetic}\overbrace {-{G\cdot M \over a}} ^{Potential}={\frac {1}{2}}{\dot {a}}^{2}-{4\pi \over 3}G\rho a^{2}.\,\!$

We define $k\equiv -{2E \over c^{2}}$, and we will show later that $k$ is a measure of the curvature of the universe:

$k{\begin{cases}>0&\,for\ E<0\ (bound)\\=0&\,for\ E=0\ (critical)\\<0&\,for\ E>0\ (unbound)\\\end{cases}}\,\!$

where $k$ has units of ${1 \over length^{2}}$ if $a$ is dimensionless. Substituting $k$ into the above energy equation, and solving for ${{\dot {a}} \over a}$, we get:

${H^{2}=\left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\rho -{kc^{2} \over a^{2}}}\,\!$

($1^{st}$ Friedmann Equation) This is a statement of conservation of $E$. The first law of thermodynamics ($\Delta S=0$) requires that any system with positive pressure must lose energy as the volume enclosing it expands. Thus, if $U$ is our internal energy and $P$ is our pressure:

${dU \over dt}=-P{dV \over dt}.\,\!$

In an expanding universe, $U={E \over V}\cdot V=\rho a^{3}$, where $\rho$ is the energy density of the universe. $P$ is the pressure of the photon gas, so:

${\dot {\rho }}a^{3}+3\rho a^{2}{\dot {a}}=-P3a^{2}{\dot {a}},\,\!$

which simplifies to:

${{\dot {\rho }}=-3{{\dot {a}} \over a}(\rho +P)}\,\!$

($2^{nd}$ Friedmann Equation) This is a a statement of the temperature loss of the universe due to adiabatic expansion.

Finally, ${\frac {d}{dt}}$($1^{st}$ Friedmann Equation) gives us:

$2{\dot {a}}{\ddot {a}}={8\pi \over 3}G{d \over dt}(\rho a^{2})={8\pi \over 3}Ga^{2}({\dot {\rho }}+2{{\dot {a}} \over a}\rho ).\,\!$

Substituting the $2^{nd}$ Friedmann Equation for ${\dot {\rho }}$:

$2{\dot {a}}{\ddot {a}}={8\pi \over 3}Ga^{2}(-{{\dot {a}} \over a}\rho -3{{\dot {a}} \over a}P)=-{8\pi \over 3}G{{\dot {a}} \over a}(\rho +3P).\,\!$

Now we have our $3^{rd}$ Equation:

${{{\ddot {a}} \over a}=-{4\pi G \over 3}(\rho +3P)}\,\!$

($3^{rd}$ Friedmann Equation) The $3^{rd}$ Friedmann Equation relates the acceleration of the expansion of the universe to the pressure of photon gas and the density of the universe. Note that if $3P\leq -\rho$, we have an accelerating universe.

Compare the $3^{rd}$ Friedmann Equation to the Newtonian equation for gravity, with $M={4\pi \over 3}\rho x^{3}$ and $\rho _{eff}=\rho +3P$:

${\ddot {x}}={-G\cdot M \over x^{2}}={-4\pi \over 3}G\rho x\,\!$